Question 1.1.1: Derive the equation of motion of the pendulum in Figure 1.1....
Derive the equation of motion of the pendulum in Figure 1.1.
Consider the schematic of a pendulum in Figure 1.1(a). In this case, the mass of the rod will be ignored as well as any friction in the hinge. Typically, one starts with a photograph or sketch of the part or structure of interest and is immediately faced with having to make assumptions. This is the “art” or experience side of vibration analysis and modeling. The general philosophy is to start with the simplest model possible (hence, here we ignore friction and the mass of the rod and assume the motion remains in a plane) and try to answer the relevant engineering questions. If the simple model doesn’t agree with the experiment, then make it more complex by relaxing the assumptions until the model successfully predicts physical observation. With the assumptions in mind, the next step is to create a free-body diagram of the system, as indicated in Figure 1.1(b), in order to identify all of the relevant forces. With all the modeled forces identified, Newton’s second law and Euler’s second law are used to derive the equations of motion.
In this example Euler’s second law takes the form of summing moments about point O. This yields
\sum{M_{O}} = Jαwhere M_{O} denotes moments about the point O, J = ml² is the mass moment of inertia of the mass m about the point O, l is the length of the massless rod, and α is the angular acceleration vector. Since the problem is really in one dimension, the vector sum of moments equation becomes the single scalar equation
Jα(t) = -mgl \sin θ(t) or ml²\ddot{θ}(t) + mgl \sin θ(t) = 0
Here the moment arm for the force mg is the horizontal distance l \sin θ, and the two overdots indicate two differentiations with respect to the time, t. This is a second-order ordinary differential equation, which governs the time response of the pendulum. This is exactly the procedure used in the first course in dynamics to obtain equations of motion.
The equation of motion is nonlinear because of the appearance of the \sin θ and hence difficult to solve. The nonlinear term can be made linear by approximating the sine for small values of θ(t) as \sin θ ≈ θ. Then the equation of motion becomes
\ddot{θ}(t) + \frac{g}{l}θ(t) = 0
This is a linear, second-order ordinary differential equation with constant coefficients and is commonly solved in the first course of differential equations (usually the third course in the calculus sequence). As we will see later in this chapter, this linear equation of motion and its solution predict the period of oscillation for a simple pendulum quite accurately. The last section of this chapter revisits the nonlinear version of the pendulum equation.